Analytic derivatives of the matrix exponential for estimation of linear continuous-time models

Linear-in-variables continuous-time processes are estimated nonlinearly, because the coefficients of the implied linear-in-variables discrete-time estimating equations are the exponential of a matrix formed with the continuous-time parameters. Even with sampling complications such as irregular intervals, mixed frequencies, and stock and flow variables, using Van loan's (1978) results, the mapping from continuous- to discrete-time parameters and its derivatives can be expressed as the submatrix of a matrix exponential. For quicker estimation and more accurate hypothesis testing or sensitivity analysis, it is often better to compute analytically the first-order derivatives of the mapping. This paper explains how to compute efficiently the continuous- to discrete-time parameter mapping and its derivatives, without computing an eigenvalue decomposition, the common way of doing this. By linking present results with previous ones, a complete chain rule is obtained for computing the Gaussian likelihood function and its derivatives with respect to the continuous-time parameters.     

On the computational competitiveness of full-information maximum-likelihood and three-stage least-squares in the estimation of nonlinear,simultaneous-equations models

While full-information maximum-likehood (FIML) estimation has long been considered an important theoretical econometric estimation technique, computational considerations have greatly restricted its use in practice. Recent advances in numerical analysis and in computational software, however, have combined to provide algorithms capable of carrying out the FIML calculations quite efficiently relative to past standards. This paper compares the computational competitiveness of FIML with its most popular competitor, 3SLS, in the estimation of a variety of linear and non-linear (in parameters and variables) models. The nonlinear full-information maximum-likelihood (NLFIML) estimator is described and a computatíonally efficient approximation, TRUNFIML, is defined. Nonlinear three-stage least-squares (NL3SLS) is accomplished by the method of Jorgenson-Laffont. Comparisons are made on the basis of numbers of iterations to convergence, number of function evaluations, and total computer CPU time required, this latter figure being most relevant to a comparison of computational effort and cost.     

A remark on the recursive estimation of ARMA order

Summary It is well known how, for an ARMA process of order (p ₀,q ₀), max (p ₀,q ₀) may be recursively estimatedHannan/Rissanen. Assuming max (p ₀,q ₀) to be known and, in addition,p ₀q ₀, a simple procedure for the recursive estimation of (p ₀,q ₀) is presented.     

The invertibility of sampled and aggregated ARMA models

Summary Necessary and sufficient conditions for a sampled (resp. aggregated) stationary ARMA process to be invertible are derived. It is shown that a sampled (resp. aggregated) invertible stationary ARMA process is always invertible.     

Fully Bayesian analysis of ARMA time series models

Statistical analysis of autoregressive-moving average (ARMA) models is an important non-standard problem. No classical approach is widely accepted; legitimacy for most classical approaches is based solely on asymptotic grounds, while small sample sizes are common. The only obstacle to the Bayesian approach are designing a structure through which prior information can be incorporated and designing a practical computational method. The objective of this work is to overcome these two obstacles. In addition to the standard results, the Bayesian approach gives a different method of determining the order of the ARMA model, that is (p, q).     

FIML estimation of the dynamic simultaneous equations model with ARMA disturbances

A method is presented for the estimation of the parameters in the dynamic simultaneous equations model with vector autoregressive moving average disturbances. The estimation procedure is derived from the full information maximum likelihood approach and is based on Newton-Raphson techniques applied to the likelihood equations. The resulting two-step Newton-Raphson procedure involves only generalized instrumental variables estimation in the second step. This procedure also serves as the basis for an iterative scheme to solve the normal equations and obtain the maximum likelihood estimates of the conditional likelihood function. A nine-equation variant of the quarterly forecasting model of the US economy developed by Fair is then used as a realistic example to illustrate the estimation procedure described in the paper.     

Robust estimation for the order of finite mixture models

In this paper, we study a robust and efficient estimation procedure for the order of finite mixture models based on the minimizing a penalized density power divergence estimator. For this task, we use the locally conic parametrization approach developed by Dacunha-Castelle and Gassiate (ESAIM Probab Stat 285–317, 1997a; Ann Stat 27:1178–1209, 1999), and verify that the minimizing a penalized density power divergence estimator is consistent. Simulation results are provided for illustration.     

Large-sample estimation strategies for eigenvalues of a Wishart matrix

The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.     

Non-parametric, unconditional quantile estimation for efficiency analysis with an application to Federal Reserve check processing operations

This paper examines the technical efficiency of US Federal Reserve check processing offices over 1980–2003. We extend results from Park et al. [Park, B., Simar, L., Weiner, C., 2000. FDH efficiency scores from a stochastic point of view. Econometric Theory 16, 855–877] and Daouia and Simar [Daouia, A., Simar, L., 2007. Nonparametric efficiency analysis: a multivariate conditional quantile approach. Journal of Econometrics 140, 375–400] to develop an unconditional, hyperbolic, α-quantile estimator of efficiency. Our new estimator is fully non-parametric and robust with respect to outliers; when used to estimate distance to quantiles lying close to the full frontier, it is strongly consistent and converges at rate root-n, thus avoiding the curse of dimensionality that plagues data envelopment analysis (DEA) estimators. Our methods could be used by policymakers to compare inefficiency levels across offices or by managers of individual offices to identify peer offices.     

An exact test for a column of the covariance matrix based on a single observation

In this paper, we derive an exact test for a column of the covariance matrix. The test statistic is calculated by using a single observation. The exact distributions of the test statistic are derived under both the null and alternative hypotheses. We also obtain an analytical expression of the power function of the test for the equality of a column of the covariance matrix to a given vector. It is shown that the information contained in a single vector is large enough to ensure a good performance of the test. Moreover, the suggested test can be applied for time-dependent multivariate Gaussian processes.     

Robust estimation of ARIMA models

The paper discusses methods of estimating univariate ARIMA models with outliers. The approach calls for a state vector representation of a time-series model, on which we can then operate on using the Kalman filter. One of the additional advantages of Kalman filter operating on the state vector representation is that the method and code could easily be adapted to be applicable to the ARIMA model with missing observations. The paper investigates ways to calculate robust initial estimation of the parameters of the ARIMA model. The method proposed is based on the results obtained by R.D. Martin (1980).     

Kernel estimation for disequilibrium models for floorspace efficiency in retailing

Conclusions In this paper we have proposed new techniques for simplifying the estimation of disequilibrium models by avoiding constrained maximum likelihood methods (which cannot avoid numerous theoretical and practical difficulties mentioned above) including an unrealistic assumption of the independence of errors in demand and supply system of equations. In the proposed first stage, one estimates the relative magnitude of the residuals from the demand and supply equations nonparametrically, even though they suffer from omitted variables bias, because the coefficient of the omitted variable is known to be the same in both equations. The reason for using nonparametric methods is that they do not depend on parametric functional forms of biased (bent inward) demand and supply equations. The first stage compares the absolute values of residuals from conditional expectations in order to classify the data points as belonging to the demand or the supply curve. We estimate the economically meaningful scale elasticity and distribution parameters at the second stage from classified (separated) data.We extend nonparametric kernel estimation to the r = 4 case to improve the speed of convergence, as predicted by Singh's [1981] theory. In the first stage, r = 4 results give generally improved R² and ¦t¦ values in our study of the Dutch data—used by many authors concerned with the estimation of floorspace productivity. We find that one can obtain reasonable results by our approximate but simpler two stage methods. Detailed results are reported for four types of Dutch retail establishments. More research is needed to gain further experience and to extend the methodology to other disequilibrium models and other productivity estimation problems.This paper was processed by W. Eichhorn.     

On the efficient estimation methods for the macro-economic models nonlinear in variables

All the macro-economic models have the nonlinearity in variables within their simultaneous equations systems. I propose a full information estimation method for such models. The method is (i) asymptotically efficient, (ii) feasible in the contemporary computer technology as it consists of calculations very much like the nonlinear multipliers, and (iii) hopefully applicable to the undersized sample case which prevails in the macro-economic model building. Though two other methods are also investigated, one is found to be asymptotically inefficient, and another turns out to be inapplicable to the undersized sample case.     

Instrumental variable estimation of nonseparable models

There are many environments where knowledge of a structural relationship is required to answer questions of interest. Also, nonseparability of a structural disturbance is a key feature of many models. Here, we consider nonparametric identification and estimation of a model that is monotonic in a nonseparable scalar disturbance, which disturbance is independent of instruments. This model leads to conditional quantile restrictions. We give local identification conditions for the structural equations from those quantile restrictions. We find that a modified completeness condition is sufficient for local identification. We also consider estimation via a nonparametric minimum distance estimator. The estimator minimizes the sum of squares of predicted values from a nonparametric regression of the quantile residual on the instruments. We show consistency of this estimator.     

Recursive estimation of simultaneous equation models

A recursive instrumental variable estimator is derived. For simultaneous equation estimation, the choice of the instruments is discussed. A computationally simple and asymptotically efficient recursive estimator is proposed in this context.     

Density estimation for nonlinear parametric models with conditional heteroscedasticity

This article studies density and parameter estimation problems for nonlinear parametric models with conditional heteroscedasticity. We propose a simple density estimate that is particularly useful for studying the stationary density of nonlinear time series models. Under a general dependence structure, we establish the root n$n$ consistency of the proposed density estimate. For parameter estimation, a Bahadur type representation is obtained for the conditional maximum likelihood estimate. The parameter estimate is shown to be asymptotically efficient in the sense that its limiting variance attains the Cramér–Rao lower bound. The performance of our density estimate is studied by simulations.     

Jackknife estimation of stationary autoregressive models

This paper explores the properties of jackknife methods of estimation in stationary autoregressive models. Some general results concerning the correct weights for bias reduction under various sampling schemes are provided and the asymptotic properties of a jackknife estimator based on non-overlapping sub-samples are derived for the case of a stationary autoregression of order p$p$ when the number of sub-samples is either fixed or increases with the sample size at an appropriate rate. The results of a detailed investigation into the finite sample properties of various jackknife and alternative estimators are reported and it is found that the jackknife can deliver substantial reductions in bias in autoregressive models. This finding is robust to departures from normality, ARCH effects and misspecification. The median-unbiasedness and mean squared error properties are also investigated and compared with alternative methods as are the coverage rates of jackknife-based confidence intervals.     

Realized Laplace transforms for estimation of jump diffusive volatility models

We develop an efficient and analytically tractable method for estimation of parametric volatility models that is robust to price-level jumps. The method entails first integrating intra-day data into the Realized Laplace Transform of volatility, which is a model-free estimate of the daily integrated empirical Laplace transform of the unobservable volatility. The estimation is then done by matching moments of the integrated joint Laplace transform with those implied by the parametric volatility model. In the empirical application, the best fitting volatility model is a non-diffusive two-factor model where low activity jumps drive its persistent component and more active jumps drive the transient one.     

Asymptotically efficient recursive estimation for incomplete data models using the observed information

For a recursive maximum-likelihood estimator with step lengths decaying as 1/n, an adaptive matrix needs to be incorporated to obtain asymptotic efficiency. Ideally, this matrix should be chosen as the inverse Fisher information matrix, which is usually very difficult to compute for incomplete data models. In this paper we give conditions under which the observed information can be incorporated into the recursive procedure to yield an efficient estimator, and we also investigate the finite sample properties of these estimators by simulation.